Proportional Segments

7/28/2019 Proportional Segments

http://slidepdf.com/reader/full/proportional-segments 1/30

PROPORTIONAL SEGMENTS&

BASIC SIMILARITY THEOREM



ILLUSTRATION

Pt. X dividessegment ABso that AX to

XB is 3 : 2. Pt. Y divides

segment CD

so that CY to YD is 3 : 2

A

D YC

X B12 8

6 4



THEOREM: PROPORTIONAL SEGMENTS “Two segments are divided proportionally if

the measures of the segments of one have

the same ratio as the measures of thecorresponding segments of the other.”

A

D YC

X B12 8

6 4 AX CY

12

XB YD

68 4

SOME PROPORTIONS

3 3

2 2

1.






A

D YC

X B12 8

6 4 AB CD

20

XB YD

10

8 4

SOME PROPORTIONS

5 5

2 2

2.






A

D YC

X B12 8

6 4 AB CD

20

AX CY

10

12 6

SOME PROPORTIONS

5 5

3 3

3.






A

D YC

X B12 8

6 4 AX +AB CY +CD

32

AX CY

16

12 6

SOME PROPORTIONS

8 8

3 3

4.



Illustrative Examples



Suppose segment AC and segment MP are

divided proportionally by points B and N

respectively. Then,

A

PNM

B C8 12

2 3 AB MN

AB

BC NP

BCMN NP

1.

2.



Suppose segment AC and segment MP are

divided proportionally by points B and N

respectively. Then,

A

PNM

B C8 12

2 3 AB MN

BC

AC MP

NP AC MP

3.

4.



Find the unknown parts assuming the

segments are divided proportionally.

X 32

6 8

Solution:

X : 32 = 6 : 8

Applying the law of proportion

8(x) = 6( 32)

8x = 192X = 24



Illustrative examples

SOLVE:

One string is divided into lengths 18 cm

and 15 cm. a second string is also to be

divided into such that the two strings will

become proportional. If the longest portion

of the second string has length 60 cm,

what is the length of the other portion of the second string?



Solution:

Let x = the length of the other portion of the second string.

x 60

15 18

x18

60

60

15

5 x

66x 5(60)

X = 50, the length of the other portionof the second string

6x = 300



BASIC PROPORTIONALITY

THEOREM If a line intersects

two sides of a

triangle and isparallel to the thirdside, then it divides

the first two sidesproportionally.



RESTATEMENT OF THE

THEOREM If a line (EF) intersects

two sides ( AB & CB) of

a triangle ( ABC) andis parallel to the thirdside( AC ), then it

divides the first twosides proportionally.

Thus,

B

E F

CA



OTHER PROPORTIONS

1. BE : EA = BF : FC

2. BE : BA = BF : BC

3. BA : EA = BC : FC 4. BE : BF = EA : FC

5. FC : EA = BC : BA

6. EF : AC = BF : BC

7. EF : AC = BE : BA

B

E F

CA



VERIFYING A PROPORTIONS( an example)

1. BE : EA = BF : FC

15 : 5 = 12 : 4

By simplifying, 3 : 1 = 3 : 1

B

E F

CA

15

4

12

58

6



VERIFYING A PROPORTIONS

2. BE : BA = BF : BC

15 : 20 = 12 : 16

By simplifying,

3 : 4 = 3 : 4

B

E F

CA

15

4

12

58

6




3. BA : EA = BC : FC

20 : 5 = 16 : 4

By simplifying,

4 : 1 = 4 : 1

B

E F

CA

15

4

12

58

6




4. BE : BF = EA : FC

15 : 12 = 5 : 4

By simplifying,

5 : 4 = 5 : 4

B

E F

CA

15

4

12

58

6




5. FC : EA = BC : BA

4 : 5 = 16 : 20

By simplifying,

4 : 5 = 4 : 5

B

E F

CA

15

4

12

58

6




6. EF : AC = BF : BC

6 : 8 = 12 : 16

By simplifying,

3 : 4 = 3 : 4

B

E F

CA

15

4

12

58

6




6. EF : AC = BE : BA

6 : 8 = 15 : 20

By simplifying,

3 : 4 = 3 : 4

B

E F

CA

15

4

12

58

6



Exercises

GIVEN: DE // BC,

AD = 9, AE = 12,

DE = 10,DB = 18.Find,

BC, AC and CE.

A

D E

CB

912

18

10



Solution

Find BC,BC : DE = BA : DA

BC : 10 = 27 : 9

orBC : 10 = 3 : 1

Applying principleof proportion

BC(1) = 10(3)

BC = 30

A

D E

CB

912

18

10

30



Solution

Find AC,AC : AE = BA : DA

AC : 12 = 27 : 9

orAC : 12 = 3 : 1


AC(1) = 12(3)

AC = 36

A

D E

CB

912

18

10



Solution

Find CE,CE : AE = BD : DA

CE : 12 = 18 : 9

orCE : 12 = 2 : 1


CE(1) = 12(2)

CE = 24

A

D E

CB

912

18

10

24



Solution

Another way tofind CE,

CE = AC - AE

Hence, AC =36,then

CE = 36 - 12

CE = 24

A

D E

CB

912

18

10

24



Quiz

Solve the following problem. Show your

solution.( one –half crosswise)

1. Uncle Tom plans to divide an80- meter rope into three pieces in

the ratio 3 : 5 : 8. what will be the

length of each piece?



QUIZ

2. In the figure, find the values of xand y.

y

15

10

30

12

x



Assignment

Answer test yourself nos. 1 – 5 and 18 and

19

Geometry textbook, page 148

Proportional Segments

Documents

Transcript of Proportional Segments